Problem 57 - Calculus Early Transcendentals

Question

If $$f(x)=\left\\{\begin{array}{ll}{x^{2}} & {\text { if } x \text { is rational }} \\ {0} & {\text { if } x \text { is irrational }}\end{array}\right.$$ prove that \(\lim _{x \rightarrow 0} f(x)=0.\)

Step-by-Step Solution

Answer

The limit of \( f(x) \) as \( x \to 0 \) is 0 by showing \( |f(x) - 0| < \epsilon \) for any \( \epsilon > 0 \).

Step 1: Understanding the Function

First, we need to understand the function \( f(x) \). It assigns values based on whether \( x \) is rational or irrational. For a rational \( x \), \( f(x) = x^2 \). For an irrational \( x \), \( f(x) = 0 \). Our goal is to find \( \lim_{x \to 0} f(x) \), meaning the value \( f(x) \) approaches as \( x \) gets infinitely close to 0, regardless of whether \( x \) is rational or irrational.

Step 2: Approach Using Epsilon-Delta Argument

To prove \( \lim_{x \to 0} f(x) = 0 \) using the epsilon-delta definition, we need to show that for any \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that if \( |x| < \delta \), then \( |f(x) - 0| < \epsilon \). This means \( |f(x)| < \epsilon \) when \( x \) is sufficiently close to 0.

Step 3: Analyzing Rational x

For rational \( x \), \( f(x) = x^2 \). If \( |x| < \delta \), then \( |f(x)| = |x^2| = x^2 < \epsilon \). Therefore, we need to choose \( \delta \) such that \( x^2 < \epsilon \), i.e., \( \delta = \sqrt{\epsilon} \). This ensures that when \( 0 < |x| < \delta \), then \( x^2 < \epsilon \).

Step 4: Considering Irrational x

For irrational \( x \), \( f(x) = 0 \). Thus, \( |f(x)| = |0| = 0 < \epsilon \) for any \( \epsilon > 0 \). This condition is inherently satisfied for all irrational \( x \) near 0.

Step 5: Conclusion of Limit Proof

By considering both rational and irrational values of \( x \), we have shown that for any \( \epsilon > 0 \), there exists a \( \delta = \sqrt{\epsilon} \) such that \( |f(x) - 0| < \epsilon \) whenever \( 0 < |x| < \delta \). Therefore, by the epsilon-delta definition of limits, \( \lim_{x \to 0} f(x) = 0 \).

Key Concepts

Epsilon-Delta Definition

The epsilon-delta definition is a formalism used to define the limit of a function at a particular point. This definition is crucial in rigorously establishing the concept of limits in calculus. According to this definition, for a function \( f(x) \) and a limit \( L \) as \( x \rightarrow a \), the statement \( \lim_{x \to a} f(x) = L \) means:

• For every \( \epsilon > 0 \) (representing how close \( f(x) \) should be to \( L \)), there exists a \( \delta > 0 \) (representing how close \( x \) should be to \( a \)) such that whenever \( 0 < |x - a| < \delta \), it follows that \( |f(x) - L| < \epsilon \).

This definition requires that we can make \( f(x) \) as close as we want to \( L \) by making \( x \) sufficiently close to \( a \), but not equal to \( a \). It provides a way to prove limits in a precise manner. In the given example, to prove the limit \( \lim_{x \to 0} f(x) = 0 \) using this definition, we demonstrate that for every positive \( \epsilon \), we can find a corresponding \( \delta \) (specifically, \( \delta = \sqrt{\epsilon} \)) that satisfies the conditions of the definition.

Rational and Irrational Functions

In mathematics, functions may behave differently when their inputs are rational numbers versus irrational numbers. Understanding this behavior is crucial, especially when working with piecewise functions like the one given in the exercise. A rational number is any number that can be expressed as the quotient of two integers, while an irrational number cannot be expressed this way.

• For rational \( x \), the function \( f(x) = x^2 \).
• For irrational \( x \), the function \( f(x) = 0 \).

This particular function is defined separately for rational and irrational inputs, highlighting how distinct the behavior of the function can be. It's important to note this duality to correctly analyze the limit problem at hand. The choice to square rational numbers while setting the function to zero for irrationals allows us to explore the limits and continuity of such piecewise functions as \( x \) approaches zero.

Continuity of Functions

A function is continuous at a point if its limit at that point equals its value at that point. For a function \( f(x) \) to be continuous at \( x = a \), it must satisfy:

• \( \lim_{x \to a} f(x) = f(a) \)

Continuity ensures there are no abrupt changes or gaps at \( a \). In the context of the problem, the function \( f(x) \) is not necessarily continuous at all points, particularly at \( x = 0 \), because the definition varies based on whether \( x \) is rational or irrational.

However, the proof focuses on the behavior as \( x \) approaches zero, demonstrating continuity through copious examination of limits. Regardless of whether \( x \) is rational or irrational, as long as it approaches zero, the function's value approaches the same limit of zero, indicating that it can be made continuous in the sense of limits. This type of function exemplifies the nuanced approach needed to grasp continuity and dissect the conditions under which a function behaves smoothly without jumps or breaks at specific points.